Summary - Winning idea for blackjack (Wouldn't it be fun to beat the casino?)

Key observations
- About half of the cards is played before shuffle. (Generally, casino plays 6 decks and use approximately 3 decks before shuffle.)
- Players are allowed to make a different amount of bets at beginning of each round.
Intuition
- Since the decks are not shuffled every round, the odds of winning changes every round depending on the cards played.
- In fact, this game is reasonably fair. That is, for some of the rounds, player will have an advantage. (I will justify this in winning strategy)
- Betting more when it is to the player's advantage will improve the overall odds.
Sum up
- On average, the casino has an advantage. However, as more cards are revealed, sometimes player might gain a small edge. If I bet significantly more than usual when the situation is favorable, player may just be able to win

Optimal strategy - Table of Content

Note: If you only want to know how to win, read the following sections: Summary, Decision Tables, Variable Bets
Decision table for complete deck
Algorithm
- Dealer's ending total
- Expectation table
Simulation
Variable Bets
Source code. (Implemented in MS VC++ 6.0)
Summary

Decision Table (dealer has an estimated 0.72% advantage)

Notation

hit: hit
sta: stand
2/h: double if possible, otherwise hit
2/s: double if possible, otherwise stand

Rules - This is the set of rule used in all the computations below

Dealer hits soft 17
Double any two cards
No double after split
No insurance
No surrender
Player can split only once
Blackjack pays 3 to 2
6 decks, shuffle after 3 decks

Tables

The first column describes dealer's up cards
The first row shows player's current cards
For instance, in hard total, entry (2, 12) = hit, this means when dealer is showing 2 and player's current total is 12, a decision to hit yields the best expectation.

Additional notes

This set of rule is chosen out because it has worse odd for the player (out of all typical blackjack rules). Thus, if I can demonstrate that player could win under such circumstance, player should be able to win under all circumstances.
Following the below strategy, player still have a disadvantage. On average, player estimated lost 0.7241 percent for every game played.

Source code
- Here is the code that generates the table below, the code works on any set of cards. (i.e. one could specify a deck with two missing A and 3 missing 5, etc)

Optimal Strategy for Hard Total

HARD	8	9	10	A	12	13	14	15	16	17
2	hit	hit	2/h	2/h	hit	sta	sta	sta	sta	sta
3	hit	2/h	2/h	2/h	hit	sta	sta	sta	sta	sta
4	hit	2/h	2/h	2/h	sta	sta	sta	sta	sta	sta
5	hit	2/h	2/h	2/h	sta	sta	sta	sta	sta	sta
6	hit	2/h	2/h	2/h	sta	sta	sta	sta	sta	sta
7	hit	hit	2/h	2/h	hit	hit	hit	hit	hit	sta
8	hit	hit	2/h	2/h	hit	hit	hit	hit	hit	sta
9	hit	hit	2/h	2/h	hit	hit	hit	hit	hit	sta
10	hit	hit	hit	2/h	hit	hit	hit	hit	hit	sta
A	hit	hit	hit	2/h	hit	hit	hit	hit	hit	sta

Optimal Strategy for Soft Total

SOFT	A2	A3	A4	A5	A6	A7	A8	A9
2	hit	hit	hit	hit	hit	2/s	sta	sta
3	hit	hit	hit	hit	2/h	2/s	sta	sta
4	hit	hit	2/h	2/h	2/h	2/s	sta	sta
5	hit	2/h	2/h	2/h	2/h	2/s	sta	sta
6	2/h	2/h	2/h	2/h	2/h	2/s	2/s	sta
7	hit	hit	hit	hit	hit	sta	sta	sta
8	hit	hit	hit	hit	hit	sta	sta	sta
9	hit	hit	hit	hit	hit	hit	sta	sta
10	hit	hit	hit	hit	hit	hit	sta	sta
A	hit	hit	hit	hit	hit	hit	sta	sta

Optimal Strategy for Decision on Split

SPLT	2	3	4	5	6	7	8	9	10	A
2	N	N	N	N	N	Y	Y	Y	N	Y
3	N	N	N	N	Y	Y	Y	Y	N	Y
4	Y	Y	N	N	Y	Y	Y	Y	N	Y
5	Y	Y	N	N	Y	Y	Y	Y	N	Y
6	Y	Y	N	N	Y	Y	Y	Y	N	Y
7	Y	Y	N	N	N	Y	Y	N	N	Y
8	N	N	N	N	N	N	Y	Y	N	Y
9	N	N	N	N	N	N	Y	Y	N	Y
10	N	N	N	N	N	N	Y	N	N	Y
A	N	N	N	N	N	N	Y	N	N	Y

Algorithm - How do I generated the table above? (Technical/Statistical)

Generation of "Dealer's ending total"
- Problem:
  - Given: Dealer's up card, and configuration of the deck
  - Output: The estimated probability that dealer will end up with 17, 18, 19, 20, 21 and so on.
- Algorithm: (Dynamic programming)
  - Notation:
    - Let d = dealer's up card
    - Let PS(n, k) = probability that dealer has a soft total of n with at most k cards.
    - Let PH(n, k) = probability that dealer has a hard total of n with at most k cards.
    - Let P(i) = probability that a random card in the deck is i. e.g. for a complete deck, P(A) = P(1) = 1/13, P(2) = 1/13, P(10) = 4/13
  - Base case:
    - if (d == Ace)
      
      PS(11, 1) = 100%
      PH( i, 1) = 0% for all i
      PS( i, 1) = 0% for all i != 11
      (Initialize PH( i, j) = 0% for all i and j > 1)
      (Initialize PS( i, j) = 0% for all i and j > 1)
  - Recursive Step/Pseudocode:
    - for k = 1 to 21 //there can be at most 21 cards in a hand, so at most 21 iterations
      
      for i = 2 to 16 //dealer only hit on hard total if the total is less than 16
      
      for j = Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10
      
      If (j != Ace) Then PH(i+j, k+1) = PH(i+j, k+1) + PH(i, k) * P(j)
      Else
      
      If (i + 11 <= 21) Then PS(i+11, k+1) = PS(i+11, k+1) + PH(i, k) * P(Ace)
      Else PH(i+1, k+1) = PH(i+1, k+1) + PH(i, k) * P(Ace)
      
      for i = 17 to 21+10 //dealer stand when more than 17 on hard total
      
      PH(i, k+1) = PH(i, k)
      
      for i = 2 to 17 //dealer hit on soft total if total is less than 17
      
      for j = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
      
      If (i + j <= 21) Then PS(i+j, k+1) = PS(i+j, k+1) + PS(i, k) * P(j)
      Else PH(i+j-10, k+1) = PH(i+j-10, k+1) + PS(i, k) * P(j)
      
      for i = 18 to 21 //dealer stand when more than 18 on soft total
      
      PS(i, k+1) = PS(i, k)
  - Result: (Probability that dealer will end up with a total of n) = PS(n) + PH(n)
- Note:
  - This algorithm is fairly accurate, but not exact.
  - It assume that P(i) are constants for each iteration.
Generation for each part of the expectation table, given the dealer's ending total, and assume the bet is 1 dollars
- First, figure out the expectation for stand and double. (This part is not particularly hard, for detail, download my source code)
- Second, figure out the expectation for hit and following the decision in the table.
  - Since we are generating the table, this step seems difficult, but it's can be done easily with a simple dynamic programming.
  - To compute the expectation for hit on a hard total of k only depends on decisions for hard totals above k, thus, one can figure out the decision of hard total starting at 21, then 20, 19, and so on.
  - Similarly, one can use this algorithm on soft total, with a simple clever trick.
- Thrid, figure out the expectation for split
  - Since we know the expectation for every card, the expectation for spliting a card with value k is simply (the expectation of value k) * 2.
  - However, if the rule allow Double after Split and Split after Split, one needs to be very careful to implement this part correctly.
- Note:
  - The purpose of computing the expectation table is to help player to achieve the best decision.
  - Thus, the fact that player is making a decision on hit/stand/double means that dealer does not have a blackjack, and the dealer's ending total should be adjusted accordingly.

Dealer's ending total -- Result only

Tables
- The rows describes dealer's up cards
- The cols shows dealer's final total
- For instance, entry (2, 18) = 0.134907, this means there are approximately 13.5% chances that dealer with an up card of 2 will end up with a total of 18.
Additional notes
- Soft 17 = a total of 17 that contains an ace that is counted as 11.
- These kinds of table is extremely important in finding out the best decision in blackjack. That is, finding the solution to the following problem.
  - Input: All the cards that has been played. Dealer's up card, player's hand.
  - Output: One of the decisions (Hit, Stand, Double, Split) that gives the best gain.
- These tables is generated assuming that no cards has been played from the deck. Thus, it is not exact.
Source code
- Here is the code that generates the table below, the code works on any set of cards. (i.e. one could specify a deck with two missing A and 3 missing 5, etc)

Dealer stands at 17 or more

Dealer's-up-card	17	18	19	20	21	Bust
2	0.139809	0.134907	0.129655	0.124027	0.117994	0.353608
3	0.135034	0.130482	0.125581	0.120329	0.114700	0.373875
4	0.130490	0.125938	0.121386	0.116485	0.111233	0.394468
5	0.122251	0.122251	0.117700	0.113148	0.108246	0.416404
6	0.165438	0.106267	0.106267	0.101715	0.097163	0.423151
7	0.368566	0.137797	0.078625	0.078625	0.074074	0.262312
8	0.128567	0.359336	0.128567	0.069395	0.069395	0.244741
9	0.119995	0.119995	0.350765	0.119995	0.060824	0.228425
10	0.111424	0.111424	0.111424	0.342194	0.111424	0.212109
A	0.130789	0.130789	0.130789	0.130789	0.361558	0.115286
Overall	0.145126	0.139497	0.133464	0.180252	0.120068	0.280000

Dealer stands at 17 or more, except hits on soft 17

Dealer's-up-card	17	18	19	20	21	Bust
2	0.130134	0.136546	0.131294	0.125665	0.119632	0.356728
3	0.126328	0.131957	0.127055	0.121803	0.116174	0.376682
4	0.122406	0.127307	0.122756	0.117854	0.112602	0.397075
5	0.118359	0.122911	0.118359	0.113807	0.108906	0.417659
6	0.114838	0.114838	0.114838	0.110286	0.105734	0.439467
7	0.368566	0.137797	0.078625	0.078625	0.074074	0.262312
8	0.128567	0.359336	0.128567	0.069395	0.069395	0.244741
9	0.119995	0.119995	0.350765	0.119995	0.060824	0.228425
10	0.111424	0.111424	0.111424	0.342194	0.111424	0.212109
A	0.057493	0.143204	0.143204	0.143204	0.373974	0.138920
Overall	0.133260	0.141507	0.135474	0.182262	0.122078	0.285419

Expectation Table

Rules - This is the set of rule used in all the computations below

Dealer hits soft 17
Double any two cards
No double after split
No insurance
No surrender
Player can split only once
Blackjack pays 3 to 2
6 decks, shuffle after 3 decks

Tables

The first column describes dealer's up cards
The first row shows player's current cards
For hard and soft table, there are three numbers in the entry (A;B;C)
- A -- the expectation for stand (in term of percentage of bet)
- B -- the expectation for hit and follows the table after hit
- C -- the expectation for double
For instance, in hard table, entry (6, 17) = -0.01;-0.52;-1.04, this means when dealer is showing 6 and player's current hard total is 17. If the player choose to stay, the player will lose 1% of his bet on average, if the player choose to hit, he will lose 52% of his bet on average, and if he choose to double, he will lose 104% of his bet on average.
For split table, there are two numbers in the entry (A;B)
- A -- the expectation for not split and play according to the decision tables. (in term of percentage of bet)
- B -- the expectation for split and play according to the decision tables.
For instance, in soft table, entry (6, 7;7) = -0.12;+0.06, this mean if player choose to play two 7's as a hard total of 14, player will have an expectation of -12%, on the other hand, if the player choose to split, he will have an expectation of 6%
Blue entry means a correct decision on the part of player will lead to positive expectation.

Additional notes:

These table are generated assuming the dealer does not have a blackjack.
In soft table, A10 means that the player has a soft total of 21, but NOT blackjack. A possible hand is A55.
This table is not exact, since dealer's ending total is not exact.
By computing several different expectation table, I notice that the more "big cards" is in the deck, the better it is for the player. (big card = 8, 9, 10, J, Q, K, A, etc).

Source code
- Here is the code that generates the table below, the code works on any set of cards. (i.e. one could specify a deck with two missing A and 3 missing 5, etc)

The expectation table for hard total

HARD

-0.29;-0.11;-0.57

-0.29;-0.13;-0.57

-0.29;-0.14;-0.55

-0.29;-0.11;-0.43

-0.29;-0.02;-0.21

-0.29;+0.07;+0.06

-0.29;+0.18;+0.36

-0.29;+0.24;+0.47

-0.29;-0.25;-0.51

-0.29;-0.31;-0.62

-0.29;-0.36;-0.73

-0.29;-0.42;-0.84

-0.29;-0.47;-0.95

-0.16;-0.54;-1.08

+0.11;-0.62;-1.25

+0.38;-0.73;-1.46

+0.64;-0.86;-1.71

+0.88;-1.00;-2.00

-0.25;-0.08;-0.49

-0.25;-0.09;-0.49

-0.25;-0.10;-0.47

-0.25;-0.08;-0.36

-0.25;+0.01;-0.14

-0.25;+0.10;+0.12

-0.25;+0.20;+0.41

-0.25;+0.26;+0.52

-0.25;-0.23;-0.47

-0.25;-0.29;-0.58

-0.25;-0.35;-0.70

-0.25;-0.41;-0.82

-0.25;-0.47;-0.93

-0.12;-0.53;-1.07

+0.14;-0.62;-1.24

+0.40;-0.73;-1.46

+0.65;-0.86;-1.71

+0.88;-1.00;-2.00

-0.21;-0.05;-0.41

-0.21;-0.06;-0.41

-0.21;-0.07;-0.39

-0.21;-0.04;-0.28

-0.21;+0.04;-0.07

-0.21;+0.13;+0.18

-0.21;+0.23;+0.46

-0.21;+0.28;+0.57

-0.21;-0.21;-0.43

-0.21;-0.27;-0.55

-0.21;-0.34;-0.67

-0.21;-0.40;-0.79

-0.21;-0.46;-0.92

-0.08;-0.53;-1.06

+0.17;-0.62;-1.24

+0.42;-0.73;-1.45

+0.66;-0.85;-1.71

+0.89;-1.00;-2.00

-0.16;-0.01;-0.33

-0.16;-0.02;-0.33

-0.16;-0.03;-0.31

-0.16;-0.01;-0.20

-0.16;+0.07;+0.00

-0.16;+0.16;+0.24

-0.16;+0.26;+0.51

-0.16;+0.31;+0.61

-0.16;-0.19;-0.39

-0.16;-0.26;-0.52

-0.16;-0.32;-0.64

-0.16;-0.39;-0.77

-0.16;-0.45;-0.90

-0.05;-0.52;-1.05

+0.19;-0.62;-1.23

+0.44;-0.73;-1.45

+0.67;-0.85;-1.71

+0.89;-1.00;-2.00

-0.12;+0.03;-0.24

-0.12;+0.02;-0.24

-0.12;+0.00;-0.22

-0.12;+0.03;-0.12

-0.12;+0.10;+0.08

-0.12;+0.19;+0.30

-0.12;+0.28;+0.56

-0.12;+0.33;+0.66

-0.12;-0.17;-0.34

-0.12;-0.24;-0.48

-0.12;-0.31;-0.62

-0.12;-0.38;-0.75

-0.12;-0.44;-0.89

-0.01;-0.52;-1.04

+0.22;-0.61;-1.23

+0.45;-0.73;-1.45

+0.68;-0.85;-1.71

+0.89;-1.00;-2.00

-0.48;-0.09;-0.95

-0.48;-0.12;-0.95

-0.48;-0.15;-0.89

-0.48;-0.07;-0.59

-0.48;+0.08;-0.19

-0.48;+0.17;+0.10

-0.48;+0.26;+0.39

-0.48;+0.29;+0.46

-0.48;-0.21;-0.51

-0.48;-0.27;-0.59

-0.48;-0.32;-0.67

-0.48;-0.37;-0.75

-0.48;-0.41;-0.83

-0.11;-0.48;-0.97

+0.40;-0.59;-1.18

+0.62;-0.72;-1.43

+0.77;-0.85;-1.70

+0.93;-1.00;-2.00

-0.51;-0.16;-1.02

-0.51;-0.19;-1.02

-0.51;-0.22;-1.00

-0.51;-0.21;-0.85

-0.51;-0.06;-0.45

-0.51;+0.10;-0.03

-0.51;+0.20;+0.29

-0.51;+0.23;+0.35

-0.51;-0.27;-0.62

-0.51;-0.32;-0.69

-0.51;-0.37;-0.77

-0.51;-0.42;-0.84

-0.51;-0.46;-0.92

-0.38;-0.51;-1.01

+0.11;-0.59;-1.18

+0.59;-0.71;-1.43

+0.79;-0.85;-1.70

+0.93;-1.00;-2.00

-0.54;-0.24;-1.09

-0.54;-0.27;-1.09

-0.54;-0.29;-1.07

-0.54;-0.29;-0.96

-0.54;-0.21;-0.72

-0.54;-0.05;-0.30

-0.54;+0.12;+0.14

-0.54;+0.16;+0.23

-0.54;-0.34;-0.74

-0.54;-0.39;-0.81

-0.54;-0.43;-0.88

-0.54;-0.47;-0.95

-0.54;-0.51;-1.02

-0.42;-0.55;-1.11

-0.18;-0.62;-1.23

+0.29;-0.72;-1.43

+0.76;-0.85;-1.70

+0.94;-1.00;-2.00

-0.54;-0.29;-1.08

-0.54;-0.31;-1.08

-0.54;-0.34;-1.06

-0.54;-0.32;-0.95

-0.54;-0.25;-0.75

-0.54;-0.15;-0.47

-0.54;+0.03;-0.01

-0.54;+0.12;+0.18

-0.54;-0.38;-0.80

-0.54;-0.43;-0.87

-0.54;-0.47;-0.94

-0.54;-0.50;-1.01

-0.54;-0.54;-1.08

-0.42;-0.58;-1.17

-0.18;-0.65;-1.30

+0.06;-0.73;-1.46

+0.55;-0.85;-1.70

+0.96;-1.00;-2.00

-0.60;-0.29;-1.20

-0.60;-0.32;-1.20

-0.60;-0.34;-1.18

-0.60;-0.35;-1.09

-0.60;-0.26;-0.83

-0.60;-0.12;-0.46

-0.60;+0.03;-0.04

-0.60;+0.10;+0.10

-0.60;-0.38;-0.84

-0.60;-0.43;-0.90

-0.60;-0.47;-0.96

-0.60;-0.51;-1.02

-0.60;-0.54;-1.08

-0.52;-0.58;-1.16

-0.23;-0.64;-1.28

+0.19;-0.73;-1.46

+0.60;-0.85;-1.71

+0.90;-1.00;-2.00

The expectation table for soft total

SOFT	AA	A2	A3	A4	A5	A6	A7	A8	A9	A10
2	-0.29;+0.08;-0.07	-0.29;+0.05;-0.07	-0.29;+0.02;-0.07	-0.29;+0.00;-0.07	-0.29;-0.02;-0.07	-0.16;-0.00;-0.01	+0.11;+0.06;+0.11	+0.38;+0.12;+0.24	+0.64;+0.18;+0.36	+0.88;+0.24;+0.47
3	-0.25;+0.10;-0.00	-0.25;+0.07;-0.00	-0.25;+0.05;-0.00	-0.25;+0.03;-0.00	-0.25;+0.01;-0.00	-0.12;+0.03;+0.05	+0.14;+0.09;+0.17	+0.40;+0.15;+0.29	+0.65;+0.20;+0.41	+0.88;+0.26;+0.52
4	-0.21;+0.13;+0.06	-0.21;+0.10;+0.06	-0.21;+0.08;+0.06	-0.21;+0.06;+0.06	-0.21;+0.04;+0.06	-0.08;+0.06;+0.12	+0.17;+0.12;+0.23	+0.42;+0.17;+0.35	+0.66;+0.23;+0.46	+0.89;+0.28;+0.57
5	-0.16;+0.16;+0.13	-0.16;+0.13;+0.13	-0.16;+0.11;+0.13	-0.16;+0.09;+0.13	-0.16;+0.07;+0.13	-0.05;+0.09;+0.18	+0.19;+0.15;+0.29	+0.44;+0.20;+0.40	+0.67;+0.26;+0.51	+0.89;+0.31;+0.61
6	-0.12;+0.19;+0.20	-0.12;+0.17;+0.20	-0.12;+0.14;+0.20	-0.12;+0.13;+0.20	-0.12;+0.11;+0.20	-0.01;+0.12;+0.25	+0.22;+0.18;+0.36	+0.45;+0.23;+0.46	+0.68;+0.28;+0.56	+0.89;+0.33;+0.66
7	-0.48;+0.17;-0.18	-0.48;+0.12;-0.18	-0.48;+0.08;-0.18	-0.48;+0.04;-0.18	-0.48;-0.00;-0.18	-0.11;+0.05;-0.01	+0.40;+0.17;+0.22	+0.62;+0.22;+0.32	+0.77;+0.26;+0.39	+0.93;+0.29;+0.46
8	-0.51;+0.10;-0.31	-0.51;+0.05;-0.31	-0.51;+0.01;-0.31	-0.51;-0.03;-0.31	-0.51;-0.07;-0.31	-0.38;-0.07;-0.26	+0.11;+0.04;-0.03	+0.59;+0.15;+0.20	+0.79;+0.20;+0.29	+0.93;+0.23;+0.35
9	-0.54;+0.00;-0.46	-0.54;-0.04;-0.46	-0.54;-0.08;-0.46	-0.54;-0.11;-0.46	-0.54;-0.15;-0.46	-0.42;-0.15;-0.40	-0.18;-0.10;-0.29	+0.29;+0.01;-0.07	+0.76;+0.12;+0.14	+0.94;+0.16;+0.23
10	-0.54;-0.07;-0.51	-0.54;-0.10;-0.51	-0.54;-0.14;-0.51	-0.54;-0.17;-0.51	-0.54;-0.21;-0.51	-0.42;-0.20;-0.46	-0.18;-0.14;-0.35	+0.06;-0.09;-0.24	+0.55;+0.03;-0.01	+0.96;+0.12;+0.18
A	-0.60;-0.06;-0.59	-0.60;-0.10;-0.59	-0.60;-0.13;-0.59	-0.60;-0.17;-0.59	-0.60;-0.20;-0.59	-0.52;-0.22;-0.55	-0.23;-0.16;-0.42	+0.19;-0.06;-0.23	+0.60;+0.03;-0.04	+0.90;+0.10;+0.10

The expectation table for split

SPLT	2,2	3,3	4,4	5,5	6,6	7,7	8,8	9,9	10,10	A,A
2	-0.11;-0.15	-0.14;-0.20	-0.02;-0.23	+0.36;-0.25	-0.25;-0.28	-0.29;-0.22	-0.29;-0.05	+0.11;+0.14	+0.64;+0.36	+0.08;+0.73
3	-0.08;-0.10	-0.10;-0.13	+0.01;-0.16	+0.41;-0.19	-0.23;-0.21	-0.25;-0.15	-0.25;+0.01	+0.14;+0.20	+0.65;+0.41	+0.10;+0.77
4	-0.05;-0.04	-0.07;-0.07	+0.04;-0.09	+0.46;-0.12	-0.21;-0.14	-0.21;-0.09	-0.21;+0.07	+0.17;+0.26	+0.66;+0.46	+0.13;+0.81
5	-0.01;+0.03	-0.03;+0.00	+0.07;-0.02	+0.51;-0.05	-0.16;-0.07	-0.16;-0.01	-0.16;+0.14	+0.19;+0.31	+0.67;+0.51	+0.16;+0.85
6	+0.03;+0.10	+0.00;+0.08	+0.10;+0.05	+0.56;+0.03	-0.12;+0.01	-0.12;+0.06	-0.12;+0.21	+0.22;+0.37	+0.68;+0.56	+0.20;+0.89
7	-0.09;-0.05	-0.15;-0.11	+0.08;-0.18	+0.39;-0.24	-0.21;-0.30	-0.32;-0.14	-0.41;+0.16	+0.40;+0.34	+0.77;+0.51	+0.17;+0.91
8	-0.16;-0.21	-0.22;-0.26	-0.06;-0.32	+0.29;-0.38	-0.27;-0.43	-0.37;-0.42	-0.46;-0.12	+0.11;+0.20	+0.79;+0.40	+0.10;+0.80
9	-0.24;-0.38	-0.29;-0.43	-0.21;-0.48	+0.14;-0.53	-0.34;-0.59	-0.43;-0.57	-0.51;-0.42	-0.18;-0.10	+0.76;+0.23	+0.00;+0.64
10	-0.29;-0.48	-0.34;-0.53	-0.25;-0.58	+0.03;-0.63	-0.38;-0.68	-0.47;-0.64	-0.54;-0.50	-0.18;-0.31	+0.55;+0.05	-0.07;+0.53
A	-0.29;-0.49	-0.34;-0.54	-0.26;-0.58	+0.03;-0.63	-0.38;-0.68	-0.47;-0.70	-0.54;-0.53	-0.23;-0.25	+0.60;+0.07	-0.06;+0.52

Simulation (without changing bet)

Rules - This is the set of rule used in all the computations below
- Dealer hits soft 17
- Double any two cards
- No double after split
- No insurance
- No surrender
- Player can split only once
- Blackjack pays 3 to 2
- 6 decks, shuffle after 3 decks
Simulation using only the table described above. (bet 1 dollar per round)
- For 0.25 million games. Lost 1680 dollars, i.e. -0.67% per round.
- For 0.50 million games. Lost 3843 dollars, i.e. -0.77% per round.
- For 1.00 million games. Lost 5880 dollars, i.e. -0.59% per round.
Simulation by computing the optimal decision every round, i.e. remember every card that is played and use it to decide hit/stand/double. (bet 1 dollar per round)
- For 0.25 million games. Lost 1564 dollars, i.e. -0.63% per round.
- For 0.50 million games. Lost 1986 dollars, i.e. -0.40% per round.
- For 1.00 million games. Lost 5153 dollars, i.e. -0.52% per round.
Notes
- First, without changing the bet, player always lose! Yes, even if the player remember every single card that is played, he still lose.
- Second, from the data above, it is not hard to observe that the result varies a lot. That is, variance is big. (When I have time, I will collect some bootstrap samples and estimate the variance.)
- Third, read the next section to figure out how a player can win by changing his bet.

Variable bet

Basic idea - Player bets more when it's to the player's advantage. (When Count >= 2, player has an advantage.)
Key observation - As the probility of "high cards" (A, K, Q, J, 10, 9) in the deck increase, the chance for player to win increase.
- Thus, player should keep track of the probility of high cards that is out, and change the bet accordingly.
- In facts, player only needs to know the difference between number of high cards and low cards (2, 3, 4, 5, 6, 7), and estimate the number of card left by eye.
- To sum up, here is how a player can figure out the count.
  - #x = number of card x that is out.
  - #high = #A + #K + #Q + #J + #10 + #9
  - #low = #2 + #3 + #4 + #5 + #6 + #7
  - nREMAIN = the number of cards remaining in the deck
  - nDECK = the number of deck remaining = nREMAIN / 52
  - COUNT = (#low - #high) / (nDECK)
- Note: the higher the COUNT is, the better it is for the player.
Simulation:
- This is the result of 1000 blackjack game, break down into categories by the COUNT
- The first column is floor(COUNT)
- The second column is the resulting gain/lost given the count
- The third column describes how often this count appear
- The fourth column is the average gain/lost = "Bet = 1"/"Occurance"
- The fifth column is the bet made based on the count. (bet 1 dollar for COUNT <=1, and COUNT dollars when count >= 2)
- The sixth column is the result if player bet basd on the count

COUNT	Result (Bet = 1)	Occurance	Average	Bet	Result (Bet Vary)
-4	-7.0	17	-0.41	1	-7.0
-3	-6.0	29	-0.21	1	-6.0
-2	-4.5	47	-0.10	1	-4.5
-1	2.5	125	0.02	1	2.5
0	6.0	518	0.01	1	6.0
1	-0.5	147	-0.00	1	-0.5
2	0.5	68	0.01	2	1.0
3	-9.0	32	-0.28	3	-27.0
4	3.0	6	0.50	4	12.0
5	-1.0	3	-0.33	5	-5.0
6	6.0	6	1.00	6	36.0
7	1.0	2	0.50	7	7.0
Sum	-9.0	1000	-0.09		14.5

Note:
- When the COUNT is high, the result tends to be better for the players.
- In the simulation, if player always bet 1 dollar, player will have a net lost of 9 dollars or 0.9%.
- However, if player vary his bet, he will have a net gain of 14.5 dollars, or 1.45%

Summary - Winning idea for blackjack (Wouldn't it be fun to beat the casino?)

Key observations

Intuition

Sum up

Optimal strategy - Table of Content

Decision Table (dealer has an estimated 0.72% advantage)

Optimal Strategy for Hard Total

Optimal Strategy for Soft Total

Optimal Strategy for Decision on Split

Algorithm - How do I generated the table above? (Technical/Statistical)

Dealer's ending total -- Result only

Dealer stands at 17 or more

Dealer stands at 17 or more, except hits on soft 17

Expectation Table

The expectation table for hard total

The expectation table for soft total

The expectation table for split

Simulation (without changing bet)

Variable bet

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